^{1}

^{2}

^{1}

^{2}

We consider equation

Consider equation

The equivalent differential system

We will use the following definitions.

A

A

We will call a period annulus associated with a central region

Respectively, a period annulus enclosing several (more than one) critical points will be called a

For example, there are four central regions and three nontrivial period annuli in the phase portrait depicted in Figure

Period annuli are the continua of periodic solutions. They can be used for constructing examples of nonlinear equations which have a prescribed number of solutions to the Dirichlet problem

Under certain conditions, period annuli of (

The Liénard equation with a quadratical term

The function

Our task in this article is to define the maximal number of nontrivial period annuli for (

We suppose that

The graph of a primitive function

The function

We discuss nontrivial period annuli in Section

The result below provides the criterium for the existence of nontrivial period annuli.

Suppose that

Then, there exists a nontrivial period annulus associated with a pair

It is evident that if

Consider, for example, (

The phase portrait for (

There are three pairs of non-neighboring points of maxima and three nontrivial period annuli exist, which are depicted in Figure

Consider a polynomial

Two non-neighboring points of maxima

Suppose

Then, the maximal possible number of regular pairs is

By induction, let

Let

Only the case (b) provides

(2) Suppose that for any sequence of

The couple

Suppose that

Otherwise, we have two possibilities:

either

In the first case, the interval

In the second case, the number of

Given number

the condition (A) is satisfied,

the primitive function

Consider the polynomial

Consider now the polynomial

Denote the maximal values of

For arbitrary even

If

The slightly modified polynomial

The graph of

This work has been supported by ERAF project no. 2010/0206/2DP/2.1.1.2.0/10/APIA/VIAA/011 and Latvian Council of Science Grant no. 09.1220.